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Consider the set of (column) vectors defined by X = {x ∈ R³ | x1 + x2 + x3 = 0, where x^T = [x1,x2,x3]^T}. Which of the following is TRUE?

1. (A) {[1,-1,0]^T,[1,0,-1]^T} is a basis for the subspace X.
2. (B) {[1,-1,0]^T,[1,0,-1]^T} is a linearly independent set, but it does not span X and therefore is not a basis of X.
3. (C) X is not a subspace of R³.
4. (D) None of the above.

Correct answer: (A) {[1,-1,0]^T,[1,0,-1]^T} is a basis for the subspace X.

Solution

The set of vectors {[1,-1,0]^T,[1,0,-1]^T} is linearly independent and spans the subspace defined by the equation x1 + x2 + x3 = 0, making it a valid basis for X.

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